44.4 Polynomial Subtraction

Polynomial Subtraction is the process of combining two polynomials by reversing the sign of every term in the second polynomial and then proceeding with addition, identifying like terms and combining their coefficients while carrying forward any terms that have no match.

Subtracted Polynomial Group Recognition

Procedure

The polynomial being subtracted is recognized as a single complete group, enclosed in parentheses, distinct from the polynomial it is being subtracted from.

Example

In $(5x^2+2x)-(x^2-3x+4)$, the second group, $x^2-3x+4$, is recognized as the polynomial being subtracted.

Subtraction as Additive Inverse Addition

Principle

Subtracting a polynomial is understood as adding the additive inverse of that polynomial, meaning every term's sign within it is reversed before the addition process proceeds.

Subtracted Polynomial Sign Reversal

Procedure

Every single term within the polynomial being subtracted has its sign reversed, positive becoming negative and negative becoming positive.

Example

Reversing every sign in $x^2-3x+4$ produces $-x^2+3x-4$.

Complete Second-Polynomial Negation

Requirement

Sign reversal is applied to every term of the second polynomial without exception; reversing only some terms while leaving others unchanged produces an incorrect result.

Confirming Completeness

Each term of the original subtracted polynomial is checked individually to confirm its sign has been reversed before proceeding to the addition step.

Matching-Term Coefficient Subtraction

Procedure

After sign reversal, the process continues exactly as in polynomial addition: terms sharing the same variable part across both polynomials are grouped, and their coefficients are added together.

Example

Combining the first polynomial, $5x^2+2x$, with the sign-reversed second polynomial, $-x^2+3x-4$, the degree-two terms combine:

Unmatched Subtracted Term Retention

Procedure

Any sign-reversed term with no matching variable part among the other polynomial's terms is carried forward into the final result exactly as it appears after its sign was reversed.

Example

The term $-4$, appearing only in the sign-reversed second polynomial, is carried forward unchanged into the final result.

Polynomial Difference Standardization

Procedure

Once every matching term has been combined and every unmatched term carried forward, the resulting expression is arranged into standard form, ordered by descending degree.

Example

Combining all steps produces $4x^2+5x-4$, already arranged correctly in standard form.

Polynomial Difference Degree Determination

Procedure

The overall degree of the final resulting polynomial is determined using the same degree rules established for any polynomial, selecting the greatest degree among the surviving nonzero terms.

Example

The final difference $4x^2+5x-4$ has an overall degree of $2$, the degree of its leading term.